Some people need to belong more than they need to be right. They have convictions, they have a desire to seek and establish what is true and what is not, but if it threatens their social standing--be it in personal relationships, in family, in friendships, or in society at large--they will stifle their convictions for the sake of being accepted rather than rejected.
Other people need to be right more than they need to belong. They have affections, they have a desire to be welcomed and loved and accepted, but if it means suppressing their deepest convictions--be it on a personal belief or conflict or a greater issue of capital-T Truth--they will risk rejection rather than relinquish their need to establish what is true.
Which path is best to follow depends, I think, upon the magnitude of the issue. Creating relational rifts over which football team is the best football team, or which way is the best way to make a sandwich, is absurd. On the other hand, suppressing one's beliefs about football teams and sandwich making in order to "keep the peace" is equally absurd.
On weightier subjects, it gets more complicated. I believe that, in general, we live in a society that values social harmony over the pursuit of truth. Be it scientific truths or social truths, people who persist in focusing on the question, "But what's true, here?" will eventually find themselves marginalized. Can't we all just get along?
Sometimes, though, it is more important to seek and establish what's true than it is to be liked and accepted.
When the question is science and material reality, it is far more important to base public and educational policy on evidence and reason than it is to accommodate any and every subjective belief, however contradictory to evidence and reason.
When the question is human rights, it is far more important to stand one's ground than it is to fit in with the crowd that would turn a blind eye and a deaf ear to oppression.
When the question is bullying and abuse, it is far more important to speak up and call out the actions by name--"that is bullying; that is abuse"--than it is to shut up and stop being so sensitive and annoying.
When you are convinced that you acted rightly, and someone else is convinced that you acted wrongly, it is far more important to bring your conviction into the light of open, reasonable, fair discussion than it is to just shut up and put up because someone's getting distressed by the disagreement. Arguments which degenerate into mutual verbal abuse are not constructive, but neither is pretending everything is all right when it is not. Stifling a conflict will not resolve the conflict.